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Molecular Quantum Mechanics
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Molecular Quantum 
Mechanics Fifth edition
Peter Atkins and Ronald Friedman
University of Oxford Indiana Purdue Fort Wayne
1
3
Great Clarendon Street, Oxford ox2 6dp
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Published in the United States
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© Peter Atkins and Ronald Friedman, 2011 
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Database right Oxford University Press (maker)
Second edition 1983
Third edition 1997
Fourth edition 2005
All rights reserved. No part of this publication may be reproduced,
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British Library Cataloguing in Publication Data
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Library of Congress Cataloging in Publication Data
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Typeset by Graphicraft Limited, Hong Kong
Printed in Italy
on acid-free paper by L.E.G.O. S.p.A.
ISBN 978–0–19–954142–3
10 9 8 7 6 5 4 3 2 1
Introduction and orientation 1
 1 The foundations of quantum mechanics 9
 Mathematical background 1 Complex numbers 35
 2 Linear motion and the harmonic oscillator 37
 Mathematical background 2 Differential equations 66
 3 Rotational motion and the hydrogen atom 69
 4 Angular momentum 99
 Mathematical background 3 Vectors 121
 5 Group theory 125
 Mathematical background 4 Matrices 166
 6 Techniques of approximation 170
 7 Atomic spectra and atomic structure 210
 8 An introduction to molecular structure 258
 9 Computational chemistry 295
10 Molecular rotations and vibrations 338
 Mathematical background 5 Fourier series and Fourier transforms 379
11 Molecular electronic transitions 382
12 The electric properties of molecules 407
13 The magnetic properties of molecules 437
 Mathematical background 6 Scalar and vector functions 474
14 Scattering theory 476
Resource section 513
Answers to selected exercises and problems 523
Index 529
Brief contents
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Detailed contentsDetailed contents
 2.2 Some general remarks on the Schrödinger equation 38
 (a) The curvature of the wavefunction 38
 (b) Qualitative solutions 39
 (c) The emergence of quantization 40
 (d) Penetration into non-classical regions 40
Translational motion 41
 2.3 Energy and momentum 41
 2.4 The significance of the coefficients 42
 2.5 The flux density 43
 2.6 Wavepackets 44
Penetration into and through barriers 44
 2.7 An infinitely thick potential wall 45
 2.8 A barrier of finite width 46
 (a) The case E < V 46
 (b) The case E > V 48
 2.9 The Eckart potential barrier 48
Particle in a box 49
 2.10 The solutions 50
 2.11 Features of the solutions 51
 2.12 The two-dimensional square well 52
 2.13 Degeneracy 53
The harmonic oscillator 54
 2.14 The solutions 55
 2.15 Properties of the solutions 57
 2.16 The classical limit 58
Further information 60
 2.1 The motion of wavepackets 60
 2.2 The harmonic oscillator: solution by factorization 61
 2.3 The harmonic oscillator: the standard solution 62
 2.4 The virial theorem 62
Mathematical background 2 Differential equations 66
MB2.1 The structure of differential equations 66
MB2.2 The solution of ordinary differential equations 66
MB2.3 The solution of partial differential equations 67
3 Rotational motion and the hydrogen atom 69
Particle on a ring 69
 3.1 The hamiltonian and the Schrödinger equation 69
 3.2 The angular momentum 70
 3.3 The shapes of the wavefunctions 71
 3.4 The classical limit 72
 3.5 The circular square well 73
 (a) The separation of variables 73
 (b) The radial solutions 73
Introduction and orientation 1
 0.1 Black-body radiation 1
 0.2 Heat capacities 2
 0.3 The photoelectric and Compton effects 3
 0.4 Atomic spectra 4
 0.5 The duality of matter 5
1 The foundations of quantum mechanics 9
Operators in quantum mechanics 9
 1.1 Linear operators 10
 1.2 Eigenfunctions and eigenvalues 10
 1.3 Representations 12
 1.4 Commutation and non-commutation 13
 1.5 The construction of operators 14
 1.6 Integrals over operators 15
 1.7 Dirac bracket and matrix notation 16
 (a) Dirac brackets 16
 (b) Matrix notation 17
 1.8 Hermitian operators 17
 (a) The definition of hermiticity 18
 (b) The consequences of hermiticity 19
The postulates of quantum mechanics 20
 1.9 States and wavefunctions 20
 1.10 The fundamental prescription 21
 1.11 The outcome of measurements 22
 1.12 The interpretation of the wavefunction 24
 1.13 The equation for the wavefunction 24
 1.14 The separation of the Schrödinger equation 25
The specification and evolution of states 26
 1.15 Simultaneous observables 27
 1.16 The uncertainty principle 28
 1.17 Consequences of the uncertainty principle 30
 1.18 The uncertainty in energy and time 31
 1.19 Time-evolution and conservation laws 31
Mathematical background 1 Complex numbers 35
MB1.1 Definitions 35
MB1.2 Polar representation 35
MB1.3 Operations 36
2 Linear motion and the harmonic oscillator 37
The characteristics of wavefunctions 37
 2.1 Constraints on the wavefunction 37
viii | DETAILED CONTENTS
Particle on a sphere 75
 3.6 The Schrödinger equation and its solution 75
 (a) The wavefunctions 77
 (b) The allowed energies 78
 3.7 The angular momentum of the particle 78
 3.8 Properties of the solutions 80
 3.9 The rigid rotor 81
 3.10 Particle in a spherical well 83
Motion in a Coulombic field 84
 3.11 The Schrödinger equation for hydrogenic atoms 84
 3.12 The separation of the relative coordinates 85
 3.13 The radial Schrödinger equation 86
 (a) The solutions close to the nucleus for l = 0 86
 (b) The solutions close to the nucleus for l ! 0 86
 (c) The complete solutions 87
 (d) The allowed energies 89
 3.14 Probabilities and the radial distribution function 89
 3.15 Atomic orbitals 90
 (a) s-orbitals 91
 (b) p-orbitals 91
 (c) d- and f-orbitals 93
 (d) The radial extent of orbitals 93
 3.16 The degeneracy of hydrogenic atoms 94
Further information 95
 3.1 The angular wavefunctions 95
 3.2 Reduced mass 95
 3.3 The radial wave equation 96
4 Angular momentum 99
The angular momentum operators 99
 4.1 The operators and their commutation relations 99
 (a) The angular momentum operators 100
 (b) The commutation relations 100
 4.2 Angular momentum observables 101
 4.3 The shift operators 102
The definition of the states 102
 4.4 The effect of the shift operators 103
 4.5 The eigenvalues of the angular momentum 104
 4.6 The matrix elements of the angular momentum 106
 4.7 The orbital angular momentum eigenfunctions 108
 4.8 Spin 110
 (a) The properties of spin 110
 (b) The matrix elements of spin operators 111
The angular momenta of composite systems 111
 4.9 The specification of coupled states 111
 4.10 The permitted values of the total angular 
momentum 112
 4.11 The vector model of coupled angular momenta 114
 4.12 The relation between schemes 115
 (a) Singlet and triplet coupled states 115
 (b) The construction of coupled states116
 (c) States of the configuration d2 117
 4.13 The coupling of several angular momenta 118
Mathematical background 3 Vectors 121
MB3.1 Definitions 121
MB3.2 Operations 121
MB3.3 The graphical representation of vector operations 122
MB3.4 Vector differentiation 123
5 Group theory 125
The symmetries of objects 125
 5.1 Symmetry operations and elements 126
 5.2 The classification of molecules 127
The calculus of symmetry 131
 5.3 The definition of a group 131
 5.4 Group multiplication tables 132
 5.5 Matrix representations 133
 5.6 The properties of matrix representations 136
 5.7 The characters of representations 138
 5.8 Characters and classes 139
 5.9 Irreducible representations 140
 5.10 The great and little orthogonality theorems 142
Reduced representations 146
 5.11 The reduction of representations 146
 5.12 Symmetry-adapted bases 147
 (a) Projection operators 148
 (b) The generation of symmetry-adapted bases 149
The symmetry properties of functions 151
 5.13 The transformation of p-orbitals 151
 5.14 The decomposition of direct-product bases 152
 5.15 Direct-product groups 154
 5.16 Vanishing integrals 156
 5.17 Symmetry and degeneracy 158
The full rotation group 159
 5.18 The generators of rotations 159
 5.19 The representation of the full rotation group 161
 5.20 Coupled angular momenta 162
Applications 163
Mathematical background 4 Matrices 166
MB4.1 Definitions 166
MB4.2 Matrix addition and multiplication 166
MB4.3 Eigenvalue equations 167
6 Techniques of approximation 170
The semiclassical approximation 170
Time-independent perturbation theory 174
 6.1 Perturbation of a two-level system 174
 6.2 Many-level systems 176
 (a) Formulation of the problem 177
 DETAILED CONTENTS | ix
 (b) The first-order correction to the energy 177
 (c) The first-order correction to the wavefunction 178
 (d) The second-order correction to the energy 180
 6.3 Comments on the perturbation expressions 181
 (a) The role of symmetry 182
 (b) The closure approximation 183
 6.4 Perturbation theory for degenerate states 185
Variation theory 187
 6.5 The Rayleigh ratio 187
 6.6 The Rayleigh–Ritz method 189
The Hellmann–Feynman theorem 191
Time-dependent perturbation theory 192
 6.7 The time-dependent behaviour of a two-level system 192
 (a) The solutions 193
 (b) The Rabi formula 195
 6.8 Many-level systems: the variation of constants 196
 (a) The general formulation 196
 (b) The effect of a slowly switched constant perturbation 198
 (c) The effect of an oscillating perturbation 199
 6.9 Transition rates to continuum states 201
 6.10 The Einstein transition probabilities 202
 6.11 Lifetime and energy uncertainty 204
Further information 206
 6.1 Electric dipole transitions 206
7 Atomic spectra and atomic structure 210
The spectrum of atomic hydrogen 210
 7.1 The energies of the transitions 210
 7.2 Selection rules 211
 (a) The Laporte selection rule 211
 (b) Constraints on Dl 212
 (c) Constraints on Dml 212
 (d) Higher-order transitions 213
 7.3 Orbital and spin magnetic moments 214
 (a) The orbital magnetic moment 214
 (b) The spin magnetic moment 215
 7.4 Spin–orbit coupling 215
 7.5 The fine-structure of spectra 217
 7.6 Term symbols and spectral details 218
 7.7 The detailed spectrum of hydrogen 219
The structure of helium 221
 7.8 The helium atom 221
 (a) Atomic units 221
 (b) The orbital approximation 222
 7.9 Excited states of helium 224
 7.10 The spectrum of helium 225
 7.11 The Pauli principle 227
Many-electron atoms 229
 7.12 Penetration and shielding 230
 7.13 Periodicity 232
 7.14 Slater atomic orbitals 233
 7.15 Slater determinants and the Condon–Slater rules 234
 7.16 Self-consistent fields 235
 (a) The Hartree–Fock equations 235
 (b) One-electron energies 237
 7.17 Restricted and unrestricted Hartree–Fock calculations 238
 7.18 Density functional procedures 239
 (a) The Thomas–Fermi method 239
 (b) The Thomas–Fermi–Dirac method 242
 7.19 Term symbols and transitions of many-electron atoms 243
 (a) Russell–Saunders coupling 243
 (b) Excluded terms 244
 (c) Selection rules 245
 7.20 Hund’s rules and Racah parameters 245
 7.21 Alternative coupling schemes 247
Atoms in external fields 248
 7.22 The normal Zeeman effect 248
 7.23 The anomalous Zeeman effect 249
 7.24 The Stark effect 251
Further information 253
 7.1 The Hartree–Fock equations 253
 7.2 Vector coupling schemes 253
 7.3 Functionals and functional derivatives 254
 7.4 Solution of the Thomas–Fermi equation 255
8 An introduction to molecular structure 258
The Born–Oppenheimer approximation 258
 8.1 The formulation of the approximation 258
 8.2 An application: the hydrogen molecule-ion 260
 (a) The molecular potential energy curves 260
 (b) The molecular orbitals 261
Molecular orbital theory 262
 8.3 Linear combinations of atomic orbitals 262
 (a) The secular determinant 263
 (b) The Coulomb integral 263
 (c) The resonance integral 265
 (d) The LCAO-MO energy levels for the hydrogen 
molecule-ion 265
 (e) The LCAO-MOs for the hydrogen molecule-ion 266
 8.4 The hydrogen molecule 266
 8.5 Configuration interaction 268
 8.6 Diatomic molecules 269
 (a) Criteria for atomic orbital overlap and bond formation 269
 (b) Homonuclear diatomic molecules 270
 (c) Heteronuclear diatomic molecules 272
Molecular orbital theory of polyatomic molecules 274
 8.7 Symmetry-adapted linear combinations 274
 (a) The H2O molecule 274
 (b) The NH3 molecule 276
 8.8 Conjugated p-systems and the Hückel approximation 276
 8.9 Ligand field theory 282
 (a) The SALCs of the octahedral complex 282
 (b) The molecular orbitals of the octahedral complex 282
 (c) The ground-state configuration: low- and high-spin 
complexes 283
 (d) Tanabe–Sugano diagrams 284
 (e) Jahn–Teller distortion 284
 (f) Metal–ligand p bonding 285
x | DETAILED CONTENTS
The band theory of solids 286
 8.10 The tight-binding approximation 286
 8.11 The Kronig–Penney model 288
 8.12 Brillouin zones 290
Further information 292
 8.1 Molecular integrals 292
9 Computational chemistry 295
The Hartree–Fock self-consistent field method 296
 9.1 The formulation of the approach 296
 9.2 The Hartree–Fock approach 297
 9.3 The Roothaan equations 298
 9.4 The selection of basis sets 302
 (a) Gaussian-type orbitals 303
 (b) The construction of contracted Gaussians 305
 (c) Calculational accuracy and the basis set 306
Electron correlation 307
 9.5 Configuration state functions 308
 9.6 Configuration interaction 309
 9.7 CI calculations 310
 9.8 Multiconfiguration methods 312
 9.9 Møller–Plesset many-body perturbation theory 313
 9.10 The coupled-cluster method 315
 (a) Formulation of the method 315
 (b) The coupled-cluster equations 315
Density functional theory 317
 9.11 The Hohenberg–Kohn existence theorem 317
 9.12 The Hohenberg–Kohn variational theorem 319
 9.13 The Kohn–Sham equations 319
 9.14 The exchange–correlation challenge 321
 (a) Local density approximations 321
 (b) More elaborate functionals 322
Gradient methods and molecular properties 323
 9.15 Energy derivatives and the Hessian matrix 324
 9.16 Analytical procedures 326
Semiempirical methods 326
 9.17 Conjugated "-electron systems 327
 (a) The Hückel approximation 327
 (b) The Pariser–Parr–Pople method 328
 9.18 General procedures 329
Molecular mechanics 332
 9.19 Force fields 332
 9.20 Quantum mechanics–molecular mechanics 333
10 Molecular rotations and vibrations 338
Spectroscopic transitions 338
10.1 Absorption and emission 338
10.2 Raman processes 339
Molecular rotation 340
10.3 Rotational energy levels 342
 (a) Symmetric rotors 342
 (b) Spherical rotors 344
 (c) Linear rotors 344
 (d) Centrifugal distortion 344
10.4 Pure rotational selection rules 345
 (a) The gross selection rule 345
 (b) The specific selection rules 345
 (c) Wavenumbers of allowed transitions 346
10.5 Rotational Raman selection rules 347
10.6 Nuclear statistics 349
 (a) The case of CO2 349
 (b) The case of H2 350
 (c) A more generalcase 352
The vibrations of diatomic molecules 353
10.7 The vibrational energy levels of diatomic molecules 353
 (a) Harmonic oscillation 353
 (b) Anharmonic oscillation 354
10.8 Vibrational selection rules 356
 (a) The gross selection rule 356
 (b) The specific selection rule 357
 (c) The effect of anharmonicities on allowed 
transitions 358
10.9 Vibration–rotation spectra of diatomic molecules 358
10.10 Vibrational Raman transitions of diatomic 
molecules 360
The vibrations of polyatomic molecules 361
10.11 Normal modes 362
 (a) Potential energy 362
 (b) Normal coordinates 363
 (c) Vibrational wavefunctions and energies 364
10.12 Vibrational and Raman selection rules for 
polyatomic molecules 365
 (a) Infrared activity 365
 (b) Raman activity 366
 (c) Group theory and molecular vibrations 366
10.13 Further effects on vibrational and rotational 
spectra 369
 (a) The effects of anharmonicity 369
 (b) Coriolis forces 372
 (c) Inversion doubling 373
Further information 374
10.1 Centrifugal distortion 374
10.2 Normal modes: an example 375
Mathematical background 5 Fourier series and 
Fourier transforms 379
MB5.1 Fourier series 379
MB5.2 Fourier transforms 380
MB5.3 The convolution theorem 381
11 Molecular electronic transitions 382
The states of diatomic molecules 382
11.1 The Hund coupling cases 382
11.2 Decoupling and L-doubling 384
11.3 Selection and correlation rules 386
 DETAILED CONTENTS | xi
Vibronic transitions 387
11.4 The Franck–Condon principle 388
11.5 The rotational structure of vibronic transitions 390
The electronic spectra of polyatomic molecules 391
11.6 Symmetry considerations 391
11.7 Chromophores 392
11.8 Vibronically allowed transitions 393
11.9 Singlet–triplet transitions 395
The fates of excited states 396
11.10 Non-radiative decay 396
11.11 Radiative decay 398
 (a) Fluorescence 398
 (b) Phosphorescence 398
Excited states and chemical reactions 399
11.12 The conservation of orbital symmetry 399
11.13 Electrocyclic reactions 399
11.14 Cycloaddition reactions 401
11.15 Photochemically induced electrocyclic reactions 402
11.16 Photochemically induced cycloaddition reactions 404
12 The electric properties of molecules 407
The response to electric fields 407
12.1 Molecular response parameters 407
12.2 The static electric polarizability 409
 (a) The mean polarizability and polarizability volume 409
 (b) The polarizability and molecular properties 411
 (c) Polarizabilities and molecular spectroscopy 412
 (d) Polarizabilities and dispersion interaction 413
 (e) Retardation effects 416
Bulk electrical properties 417
12.3 The relative permittivity and the electric susceptibility 417
 (a) Non-polar molecules 418
 (b) Polar molecules 419
12.4 Refractive index 421
 (a) The dynamic polarizability 422
 (b) The molar refractivity 424
 (c) The refractive index and dispersion 424
Optical activity 425
12.5 Circular birefringence and optical rotation 425
12.6 Magnetically induced polarization 427
12.7 Rotational strength 429
 (a) Symmetry properties 429
 (b) Optical rotatory dispersion 429
 (c) Estimation of rotational strengths 430
Further information 432
12.1 Oscillator strength 432
12.2 Sum rules 432
12.3 The Maxwell equations 433
 (a) The general form of the equations 433
 (b) The equations for fields in a vacuum 433
 (c) The propagation of fields in a polarizable medium 434
 (d) Propagation in chiral media 434
13 The magnetic properties of molecules 437
The description of magnetic fields 437
13.1 Basic concepts 437
13.2 Paramagnetism 439
13.3 The vector potential 440
 (a) The formulation of the vector potential 441
 (b) Gauge invariance 442
Magnetic perturbations 443
13.4 The perturbation hamiltonian 443
13.5 The magnetic susceptibility 444
 (a) Expressions for the susceptibility 445
 (b) Contributions to the susceptibility 446
 (c) The role of the gauge 448
13.6 The current density 449
 (a) Real wavefunctions 450
 (b) Orbitally degenerate states, zero field 450
 (c) Orbitally non-degenerate states, non-zero field 451
13.7 The diamagnetic current density 452
13.8 The paramagnetic current density 452
Magnetic resonance parameters 454
13.9 Shielding constants 454
 (a) The nuclear field 454
 (b) The hamiltonian 455
 (c) The first-order correction to the energy 455
 (d) Contributions to the shielding constant 457
13.10 The diamagnetic contribution to shielding 458
13.11 The paramagnetic contribution to shielding 459
13.12 The g-value 460
 (a) The spin hamiltonian 460
 (b) Formulating the g-value 461
13.13 Spin–spin coupling 462
13.14 Hyperfine interactions 463
 (a) Dipolar coupling 464
 (b) The Fermi contact interaction 465
 (c) The total interaction 466
13.15 Nuclear spin–spin coupling 467
 (a) The formulation of the problem 468
 (b) Coupling through a chemical bond 470
Further information 471
13.1 The hamiltonian in the presence of a magnetic field 471
13.2 The dipolar vector potential 471
Mathematical background 6 Scalar and vector 
functions 474
MB6.1 Definitions 474
MB6.2 Differentiation 474
14 Scattering theory 476
The fundamental concepts 476
14.1 The scattering matrix 476
14.2 The scattering cross-section 479
xii | DETAILED CONTENTS
Elastic scattering 480
14.3 Stationary scattering states 480
 (a) The scattering amplitude 481
 (b) The differential cross-section 482
14.4 Scattering by a central potential 483
 (a) The partial-wave stationary scattering state 483
 (b) The partial-wave equation 484
 (c) The scattering phase shift 485
 (d) The scattering matrix element 487
 (e) The scattering cross-section 489
14.5 Scattering by a spherical square well 491
 (a) The S-wave radial wavefunction and phase shift 491
 (b) Background and resonance phase shifts 492
 (c) The Breit–Wigner formula 494
 (d) The resonance contribution to the scattering matrix 
element 496
14.6 Methods of approximation 497
 (a) The WKB approximation 498
 (b) The Born approximation 499
Multichannel scattering 503
14.7 The scattering matrix for multichannel processes 504
14.8 Inelastic scattering 504
 (a) The form of the multichannel stationary 
scattering state 505
 (b) Scattering amplitude and cross-sections 505
 (c) The close-coupling approximation 506
14.9 Reactive scattering 507
14.10 The S matrix and multichannel resonances 508
Further information 509
14.1 Green’s functions 509
Resource section 513
Further reading 513
1 Character tables and direct products 516
2 Vector coupling coefficients 520
3 Wigner–Witmer rules 521
Answers to selected exercises and problems 523
Index 529
1 www.oxfordtextbooks.co.uk/orc/mqm5e/
2 Implementation of these and other techniques can be achieved by using the Students’ edition 
of Spartan software; readers can purchase a copy of this software at special discount by visiting 
www.wavefunction.com and using the discount code OUPMQM.
In this new edition we have sought to reflect the changing emphasis in the applica-
tions of molecular quantum mechanics and to make the text more accessible 
without sacrificing rigour. We describe below the key features used to achieve this 
aim. There are many new organizational and content changes throughout. All 
the artwork has been redrawn and augmented.
We have introduced and placed brief Mathematical background sections 
following the chapter where a particular mathematical technique is used for the 
first time. The Further information sections of the previous edition have either 
been incorporated into the Mathematical background sections or moved to the 
end of the chapter to which they most directly relate. New Further information
sections, such as one on the Thomas–Fermi method (Chapter 7), have also been 
introduced. Subsections have been added to the chapters to help make the mater-
ial more digestible. A lot of material has been shipped to diCerent locations to 
make the exposition more systematic, to improve its flow, or to remove diAcult
material from early parts of the text.
Problem solving is always a diAcult but important area,and we have paid 
special attention to helping students. In the chapters we have made extensive use 
of brief illustrations to provide quick and succinct examples of the use of equa-
tions, in some cases simply to establish the order of magnitude of a property and 
not leave it as an abstract entity. As in previous editions, there are numerous 
Worked examples, which require a more detailed approach; they are accom-
panied by Self-tests, which let readers test their grasp of the approach in a related 
problem. To provide a more gentle series of tests at the end of each chapter we 
have divided the questions into straightforward Exercises and more demanding 
Problems. Answers to numerical questions are given at the end of the book. A 
Student’s solutions guide provides more detailed solutions to designated Exercises
and Problems, and an Instructor’s guide provides detailed solutions to them all. 
Both guides are available in the book’s Online Resource Centre.1
One almost entirely rewritten chapter on computational chemistry (Chapter 9) 
deals in detail with density functional theory, one of the most widely used current 
techniques. We have adopted the novel pedagogical device of developing the 
theory around the H2 molecule, which though too simple to be of much profes-
sional interest has the advantage that the approach can be illustrated in explicit 
detail, so illustrating exactly what otherwise obscure computer programs are 
achieving. Computational problems that are best solved by using software are 
available on the website.1,2
We have encouraged readers to develop their understanding by using inter-
active spreadsheets on the website, which provide opportunities to explore 
numerous equations presented in the text by substituting numerical values for 
variable parameters.
Preface
www.oxfordtextbooks.co.uk/orc/mqm5e/
www.wavefunction.com
xiv | PREFACE
We have expanded discussion in numerous places in the text to provide more 
introductory material (for example, classical magnetism in Chapter 13) or a 
more systematic treatment (for example, Hund coupling cases in Chapter 11). 
The present edition also contains other significant additions, including new or 
expanded discussions of circular and spherical square wells (Chapter 3); the 
semiclassical approximation (Chapter 6); Racah parameters, Condon–Slater 
rules and atomic units (Chapter 7); Wigner–Witmer rules (Chapter 11); gauge 
invariance (Chapter 13); and reactive scattering (Chapter 14).
We are very grateful to all those who have helped in the preparation of this 
new edition, including the following, who reviewed the textbook at various 
stages along the way:
Temer Ahmadi, Villanova University
Arjun Berera, University of Edinburgh
Alexander Brown, University of Alberta
Fabio Canepa, University of Genoa
Jonathan Flynn, University of Southampton
Ian Jamie, Macquarie University
Karl Jalkanen, Curtin University of Technology
Peter Karadakov, University of York
Thomas Miller, California Institute of Technology
Alejandro Perdomo, Harvard University
Charles Trapp, University of Louisville
Donald Truhlar, University of Minnesota
We also appreciate the insights and advice of the many others, too numerous to 
name here, who oCered suggestions over the years. Finally, we wish publically to 
thank our publisher who has been invariably helpful and understanding.
PWA
RSF
0.1 Black-body radiation 1
0.2 Heat capacities 2
0.3 The photoelectric and 
Compton effects 3
0.4 Atomic spectra 4
0.5 The duality of matter 5
Introduction and orientation
There are two approaches to quantum mechanics. One is to follow the historical 
development of the theory from the fi rst indications that the whole fabric of classical 
mechanics and electrodynamics should be held in doubt to the resolution of the problem 
in the work of Planck, Einstein, Heisenberg, Schrödinger, and Dirac. The other is to stand 
back at a point late in the development of the theory and to see its underlying 
theoretical structure. The fi rst is interesting and compelling because the theory is seen 
gradually emerging from confusion and dilemma. We see experiment and intuition jointly 
determining the form of the theory and, above all, we come to appreciate the need for 
a new theory of matter. The second, more formal approach is exciting and compelling 
in a di# erent sense: there is logic and elegance in a scheme that starts from only a few 
postulates, yet reveals as their implications are unfolded, a rich, experimentally verifi able 
structure.
This book takes that latter route through the subject. However, to set the scene we 
shall take a few moments to review the steps that led to the revolutions of the early 
twentieth century, when some of the most fundamental concepts of the nature of 
matter and its behaviour were overthrown and replaced by a puzzling but powerful new 
description.
0.1 Black-body radiation
In retrospect—and as will become clear—we can now see that theoretical physics 
hovered on the edge of formulating a quantum mechanical description of matter 
as it was developed during the nineteenth century. However, it was a series of 
experimental observations that motivated the revolution. Of these observations, 
the most important historically was the study of black-body radiation, the 
radiation in thermal equilibrium with a body that absorbs and emits without 
favouring particular frequencies. A pinhole in an otherwise sealed container is 
a good approximation (Fig. 0.1).
Two characteristics of the radiation had been identified by the end of the nine-
teenth century and summarized in two laws. According to the Stefan–Boltzmann
law, the excitance, M, the power emitted divided by the area of the emitting 
region, is proportional to the fourth power of the temperature:
M = sT4 (0.1)
The Stefan–Boltzmann constant, s, is independent of the material from which 
the body is composed, and its modern value is 56.7 nW m!2 K!4. So, a region 
of area 1 cm2 of a black body at 1000 K radiates about 6 W if all frequencies 
are taken into account. Not all frequencies (or wavelengths, with l = c/n),
though, are equally represented in the radiation, and the observed peak moves 
to shorter wavelengths as the temperature is raised. According to Wien’s dis-
placement law,
2 | INTRODUCTION AND ORIENTATION
lmaxT = constant (0.2)
with the constant equal to 2.9 mm K.
One of the most challenging problems in physics at the end of the nineteenth 
century was to explain these two laws. Each one of them concentrated on finding 
an expression for the energy density E(l),the energy in a region divided by the 
volume of the region, and writing the contribution dE(l) from radiation in the 
wavelength range l to l + dl as
dE(l) = rR(l)dl (0.3)
where rR(l) is the spectral density of states at the wavelength l. Lord Rayleigh, 
with minor help from James Jeans,1 brought his formidable experience of classical 
physics to bear on the problem, and formulated the theoretical Rayleigh–Jeans
law for this quantity
rR(l) = 
8pkT
l4
(0.4)
where k is Boltzmann’s constant (k = 1.381 " 10!23 J K!1). This formula summar-
izes the failure of classical physics. Because rR(l) becomes infinite as l approaches 
zero, eqn 0.4 suggests that regardless of the temperature, there should be an 
infinite energy density at very short wavelengths. This absurd result was termed 
by Ehrenfest the ultraviolet catastrophe.
At this point, Planck made his historic contribution. His suggestion was equiva-
lent to proposing that an oscillation of the electromagnetic field of frequency n
could be excited only in steps of energy of magnitude hn, where h is a new fun-
damental constant of nature now known as Planck’s constant. According to this 
quantization of energy, the supposition that energy can be transferred only in 
discrete amounts, the oscillator can have the energies 0, hn, 2hn, . . . , and no 
other energy. Classical physics allowed a continuous variation in energy, so even 
a very high frequency oscillatorcould be excited with a very small energy: that 
was the root of the ultraviolet catastrophe since short wavelength radiation 
could be emitted at even low temperature. Quantum theory is characterized 
by discreteness in energies (and, as we shall see, of certain other properties), and 
the need for a minimum excitation energy eCectively switches oC oscillators of 
very high frequency, and hence eliminates the ultraviolet catastrophe.
When Planck implemented his suggestion, he derived what is now called the 
Planck distribution for the spectral density of a black-body radiator:
rR(l) =
8phc
l5
e!hc/lkT
1 ! e!hc/lkT
(0.5)
This expression, which is plotted in Fig. 0.2, avoids the ultraviolet catastrophe, 
and fits the observed energy distribution extraordinarily well if we take h = 6.626 
" 10!34 J s. Just as the Rayleigh–Jeans law epitomizes the failure of classical 
physics, the Planck distribution epitomizes the inception of quantum theory. It 
began the new century as well as a new era, for it was published in 1900.
0.2 Heat capacities
In 1819, science had a deceptive simplicity. The French scientists Dulong and 
Petit, for example, were able to propose their law that ‘the atoms of all simple 
Marginal comment
Using the Worksheet entitled 
Equation 0.5 on this text’s 
website, explore the 
dependence of the Planck 
distribution on the 
temperature.
1 ‘It seems to me’, said Jeans, ‘that Lord Rayleigh has introduced an unnecessary factor 8 by 
counting negative as well as positive values of his integers.’ (Phil. Mag., 91, 10 (1905).)
Fig. 0.2 The Planck distribution.
0
0
5
10
15
20
25
0.5 1 1.5 2
Wavelength, l/(hc /kT )
S
pe
ct
ra
l d
en
si
ty
,r
(l
)/8
#(
kT
)5
/(h
c)
4
Container at a
temperature T
Detected
radiation
Pinhole
Fig. 0.1 A black-body emitter 
can be simulated by a heated 
container with a pinhole in 
the wall. The electromagnetic 
radiation is reflected many times 
inside the container and reaches 
thermal equilibrium with the 
walls.
 0.3 THE PHOTOELECTRIC AND COMPTON EFFECTS | 3
bodies have exactly the same heat capacity’ of about 25 J K!1 mol!1 (in modern 
units). Dulong and Petit’s rather primitive observations, though, were done at 
room temperature, and it was unfortunate for them and for classical physics 
when measurements were extended to lower temperatures and to a wider range 
of materials. It was found that all elements had heat capacities lower than those 
predicted by Dulong and Petit’s law and that the values tended towards zero 
as T $ 0.
Dulong and Petit’s law was easy to explain in terms of classical physics by 
assuming that each atom acts as a classical oscillator in three dimensions. The 
calculation predicted that the molar isochoric (constant volume) heat capacity, 
CV,m, of a monatomic solid should be equal to 3R = 24.94 J K!1 mol!1, where R
is the gas constant (R = NAk, with NA Avogadro’s constant). That the heat 
capacities were smaller than predicted was a serious embarrassment. Einstein 
recognized the similarity between this problem and black-body radiation, for if 
each atomic oscillator required a certain minimum energy before it would actively 
oscillate and hence contribute to the heat capacity, then at low temperatures 
some would be inactive and the heat capacity would be smaller than expected. 
He applied Planck’s suggestion for electromagnetic oscillators to the material, 
atomic oscillators of the solid, and deduced the following expression:
CV,m(T) = 3RfE(T) fE(T) = !@
qE
T
· e
qE/2T
1 ! eqE/T
#
$
2
(0.6a)
where the Einstein temperature, qE, is related to the frequency of atomic oscil-
lators by qE = hn/k. The function CV,m(T)/R, which is plotted in Fig. 0.3, provides 
a reasonable fit to experimental heat capacities except at very low temperatures, 
but that can be traced to Einstein’s assumption that all the atoms oscillated with 
the same frequency. When this restriction was removed by Debye, he obtained
CV,m(T) = 3RfD(T) fD(T) = 3
AC
T
qD
DF
3
!
qD/T
0
x4ex
(ex ! 1)2
dx (0.6b)
where the Debye temperature, qD, is related to the maximum frequency of the 
oscillations that can be supported by the solid. This expression gives a very good 
fit to experimental heat capacities.
The importance of Einstein’s contribution is that it complemented Planck’s. 
Planck had shown that the energy of radiation is quantized; Einstein showed that 
matter is quantized too. Quantization appeared to be universal. Neither was able 
to justify the form that quantization took (with oscillators excitable in steps of 
hn), but that is a problem we shall solve later in the text.
0.3 The photoelectric and Compton e! ects
In those enormously productive months of 1905–6, when Einstein formulated 
not only his theory of heat capacities but also the special theory of relativity, he 
found time to make another fundamental contribution to modern physics. His 
achievement was to relate Planck’s quantum hypothesis to the phenomenon of 
the photoelectric eIect, the emission of electrons from metals when they are 
exposed to ultraviolet radiation. The puzzling features of the eCect were that 
the emission was instantaneous when the radiation was applied however low 
its intensity, but there was no emission, whatever the intensity of the radiation, 
unless its frequency exceeded a threshold value typical of each metallic element. 
It was also known that the kinetic energy of the ejected electrons varied linearly 
with the frequency of the incident radiation.
Marginal comment
Using the Worksheet entitled 
Equation 0.6a on this text’s 
website, explore the variation 
of the Einstein molar heat 
capacity with temperature for 
diIerent values of the Einstein 
temperature.
Fig. 0.3 The Einstein and Debye 
molar heat capacities. The 
symbol q denotes the Einstein 
and Debye temperatures, 
respectively. Close to T = 0 
the Debye heat capacity is 
proportional to T3.
M
ol
ar
 h
ea
t c
ap
ac
ity
, C
V
,m
/R
1
1
2
2
3
0
0 0.5 1.5
Temperature, T/q
Debye
Einstein
Marginal comment
Using the Worksheet entitled 
Equation 0.6b on this text’s 
website, explore the variation 
of the Debye molar heat 
capacity with temperature 
for diIerent values of 
the Debye temperature.
4 | INTRODUCTION AND ORIENTATION
Einstein pointed out that all the observations fell into place if the electromag-
netic field was quantized, and that it consisted of bundles of energy of magnitude 
hn. These bundles were later named photons by G.N. Lewis, and we shall use that 
term from now on. Einstein viewed the photoelectric eCect as the outcome of a 
collision between an incoming projectile, a photon of energy hn, and an electron 
buried in the metal. This picture accounts for the instantaneous character of the 
eCect, because even one photon can participate in one collision. It also accounted 
for the frequency threshold, because a minimum energy (which is normally 
denoted F and called the ‘work function’ for the metal, the analogue of the ion-
ization energy of an atom) must be supplied in a collision before photoejection 
can occur; hence, only radiation for which hn > F can be successful. The linear 
dependence of the kinetic energy, Ek, of the photoelectron on the frequency of the 
radiation is a simple consequence of the conservation of energy, which implies that
Ek = hn ! F (0.7)
If photons do have a particle-like character, then they should possess a linear 
momentum, p. The relativistic expression relating a particle’s energy to its mass 
and momentum is
E2 = m2c4 + p2c2 (0.8)
where c is the speed of light. In the case of a photon, E = hn and m = 0, so
p = hn
c
= h
l
(0.9)
This linear momentum should be detectable if radiation falls on an electron, for 
a partial transfer of momentum during the collision should appear as a change in 
wavelength of the photons. In 1923, A.H. Compton performed the experiment 
with X-rays scattered from the electrons in a graphite target, and found the 
results fitted the following formula for the shiftin wavelength, dl = lf ! li, when 
the radiation was scattered through an angle q:
dl = 2lC sin2 12 q (0.10)
where lC = h/mec is called the Compton wavelength of the electron (lC = 2.426 pm). 
This formula is derived on the supposition that a photon does indeed have a 
linear momentum h/l and that the scattering event is like a collision between two 
particles (Problem 0.12). There seems little doubt, therefore, that electromagnetic 
radiation has properties that classically would have been characteristic of particles.
The photon hypothesis seems to be a denial of the extensive accumulation 
of data that apparently provided unequivocal support for the view that electro-
magnetic radiation is wave-like. By following the implications of experiments and 
quantum concepts, we have accounted quantitatively for observations for which 
classical physics could not supply even a qualitative explanation.
0.4 Atomic spectra
There was yet another body of data that classical physics could not elucidate 
before the introduction of quantum theory. This puzzle was the observation that 
the radiation emitted by atoms was not continuous but consisted of discrete 
frequencies, or spectral lines. The visible spectrum of atomic hydrogen had a very 
simple appearance, and by 1885 J. Balmer had already noticed that their wave-
numbers, â, where â = n/c, fitted the expression
â = RH
A
C
1
22
! 1
n2
D
F (0.11)
 0.5 THE DUALITY OF MATTER | 5
where RH has come to be known as the Rydberg constant for hydrogen (RH =
1.097 " 105 cm!1) and n = 3, 4, . . . . Rydberg’s name is commemorated because he 
generalized this expression to accommodate all the transitions in atomic hydro-
gen, not just those in the visible region of the electromagnetic spectrum. Even more 
generally, the Ritz combination principle states that the frequency of any spectral 
line could be expressed as the diCerence between two quantities, or terms:
â = T1 ! T2 (0.12)
This expression strongly suggests that the energy levels of atoms are confined 
to discrete values, because a transition from one term of energy hcT1 to another 
of energy hcT2 can be expected to release a photon of energy hcâ, or hn, equal to 
the diCerence in energy between the two terms: this argument leads directly to 
the expression for the wavenumber of the spectroscopic transitions.
But why should the energy of an atom be confined to discrete values? In clas-
sical physics, all energies are permissible. The first attempt to weld together 
Planck’s quantization hypothesis and a mechanical model of an atom was made 
by Niels Bohr in 1913. By arbitrarily assuming that the angular momentum of 
an electron around a central nucleus (the picture of an atom that had emerged 
from Rutherford’s experiments in 1910) was confined to certain values, he was 
able to deduce the following expression for the permitted energy levels of an 
electron in a hydrogen atom (Problem 0.18):
En = !
me4
8h2e20
· 1
n2
n = 1,2, . . . (0.13)
where 1/m = 1/me + 1/mp and e0 is the vacuum permittivity, a fundamental con-
stant. This formula marked the first appearance in quantum mechanics of a 
quantum number, here denoted n, which identifies the state of the system and is 
used to calculate its energy. Equation 0.13 is consistent with Balmer’s formula 
(eqn 0.11) and accounted with high precision for all the transitions of hydrogen 
that were then known.
Bohr’s achievement was the union of theories of radiation and models of 
mechanics. However, it was an arbitrary union, and we now know that it is con-
ceptually untenable (for instance, it is based on the view that an electron travels 
in a circular path around the nucleus). Nevertheless, the fact that he was able to 
account quantitatively for the appearance of the spectrum of hydrogen indicated 
that quantum mechanics was central to any description of atomic phenomena 
and properties.
0.5 The duality of matter
The grand synthesis of these ideas and the demonstration of the deep links that 
exist between electromagnetic radiation and matter began with Louis de Broglie, 
who proposed on the basis of relativistic considerations that with any moving body 
there is ‘associated a wave’, and that the momentum of the body and the wave-
length are related by the de Broglie relation:
l = h
p
(0.14)
We have seen this formula already (eqn 0.9), in connection with the properties of 
photons. De Broglie proposed that it is universally applicable.
The significance of the de Broglie relation is that it summarizes a fusion of 
opposites: the momentum is a property of particles; the wavelength is a property 
of waves. This duality, the possession of properties that in classical physics are 
6 | INTRODUCTION AND ORIENTATION
characteristic of both particles and waves, is a persistent theme in the inter-
pretation of quantum mechanics. It is probably best to regard the terms ‘wave’ 
and ‘particle’ as remnants of a language based on a false (classical) model of the 
universe, and the term ‘duality’ as a late attempt to bring the language into line 
with a current (quantum mechanical) model.
The experimental results that confirmed de Broglie’s conjecture are the obser-
vation of the diCraction of electrons by the ranks of atoms in a metal crystal 
acting as a diCraction grating. Davisson and Germer, who performed this experi-
ment in 1925 using a crystal of nickel, found that the diCraction pattern was 
consistent with the electrons having a wavelength given by the de Broglie 
relation. Shortly afterwards, G.P. Thomson also succeeded in demonstrating the 
diCraction of electrons by thin films of celluloid and gold.2
If electrons—if all particles—have wave-like character, then we should expect 
there to be observational consequences. In particular, just as a wave of definite 
wavelength cannot be localized at a point, we should not expect an electron in 
a state of definite linear momentum (and hence wavelength) to be localized at 
a single point. It was pursuit of this idea that led Werner Heisenberg to his 
celebrated uncertainty principle, that it is impossible to specify the location 
and linear momentum of a particle simultaneously with arbitrary precision. In 
other words, information about location is at the expense of information about 
momentum, and vice versa. This complementarity of certain pairs of observables, 
the mutual exclusion of the specification of one property by the specification of 
another, is also a major theme of quantum mechanics, and almost an icon of the 
diCerence between it and classical mechanics, in which the specification of exact 
trajectories (positions and momenta) was a central theme.
The consummation of all this faltering progress came in 1926 when Werner 
Heisenberg and Erwin Schrödinger formulated their seemingly diCerent but 
equally successful versions of quantum mechanics. These days, we step between 
the two formalisms as the fancy takes us, for they are mathematically equivalent, 
and each one has particular advantages in diCerent types of calculation. Although 
Heisenberg’s formulation preceded Schrödinger’s by a few months, it seemed 
more abstract and was expressed in the then unfamiliar vocabulary of matrices. 
Still today it is more suited for the more formal manipulations and deductions 
of the theory, and in the following pages we shall employ it in that manner. 
Schrödinger’s formulation, which was in terms of functions and diCerential
equations, was more familiar in style but still equally revolutionary in implica-
tion. It is more suited to elementary manipulations and to the calculation of 
numerical results, and we shall employ it in that manner. You should already 
be familiar with an application of Schrödinger’s formulation in the case of the 
‘particle in a box’, a particle confined by infinite potential energy walls to a finite 
region of one-dimensional space. Keep the solution of the particle in a box prob-
lem (which we treat in detail in Chapter 2) in mind as we unroll the postulates of 
quantummechanics in Chapter 1.
‘Experiments’, said Planck, ‘are the only means of knowledge at our disposal. 
The rest is poetry, imagination.’ It is time for that imagination to unfold.
2 It has been pointed out by M. Jammer that J.J. Thomson was awarded the Nobel Prize for 
showing that the electron is a particle, and G.P. Thomson, his son, was awarded the Prize for 
showing that the electron is a wave. (See The conceptual development of quantum mechanics,
McGraw-Hill, New York (1966), p. 254.)
 PROBLEMS | 7
Exercises
*0.1 Calculate the size of the quanta involved in the 
excitation of (a) an electronic motion of period 1.0 fs, 
(b) a molecular vibration of period 10 fs, and 
(c) a pendulum of period 1.0 s.
*0.2 The peak in the Sun’s emitted energy occurs at 
about 480 nm. Estimate the temperature of its surface 
on the basis of it being regarded as a black-body 
emitter.
*0.3 An unknown metal has a specific heat capacity of 
0.91 J K!1 g!1 at room temperature. Use Dulong and 
Petit’s law to identify the metal.
*0.4 Calculate the energy of 1.00 mol photons of 
wavelength (a) 510 nm (green), (b) 100 m (radio), 
(c) 130 pm (X-ray).
*0.5 Calculate the wavelength of the radiation scattered 
through an angle of 60o when X-rays of wavelength 
25.878 pm impinge upon a graphite target.
*0.6 Calculate the speed of an electron emitted from 
a clean potassium surface (F = 2.3 eV) by light of 
wavelength (a) 300 nm, (b) 600 nm.
*0.7 Compute the highest and lowest wavenumbers of 
the spectral lines in the Balmer series for atomic hydrogen. 
What are the corresponding wavelengths?
*0.8 Compute the energies (in joules and electronvolts) for 
the two lowest energy levels of an electron in a hydrogen 
atom.
*0.9 Calculate the de Broglie wavelength of a tennis ball of 
mass 57 g travelling at 80 km h!1.
Problems
*0.1 Find the wavelength corresponding to the maximum in 
the Planck distribution for a given temperature, and show 
that the expression reduces to the Wien displacement law at 
short wavelengths. Determine an expression for the constant 
in the law in terms of fundamental constants. (This constant 
is called the second radiation constant, c2.)
0.2 Show that the Planck distribution reduces to the 
Rayleigh–Jeans law at long wavelengths.
0.3 Compute the power emitted by the Sun regarding it as 
a black-body radiator at 6 kK; the Sun has a surface area 
of 6 " 1018 m2. What energy is emitted during a 24-hour 
period?
*0.4 Derive the Einstein formula for the heat capacity of 
a collection of harmonic oscillators. To do so, use the 
quantum mechanical result that the energy of a harmonic 
oscillator of force constant kf and mass m is one of the 
values (n + 1/2)hn with n = (1/2p)(kf /m)1/2 and n = 0, 1, 
2, . . . . Hint. Calculate the mean energy, E, of a collection 
of oscillators by substituting these energies into the 
Boltzmann distribution, and then evaluate C = dE/dT.
0.5 Find the (a) low temperature, (b) high temperature 
forms of the Einstein heat capacity function.
0.6 Show that the Debye expression for the heat capacity 
is proportional to T3 as T $ 0.
*0.7 Estimate the molar heat capacities of metallic sodium 
(qD = 150 K) and diamond (qD = 1860 K) at room 
temperature (300 K).
0.8 Calculate the molar entropy of an Einstein solid at
T = qE. Hint. The entropy is S(T) = 2T0 (CV/T)dT. Evaluate 
the integral numerically.
0.9 How many photons would be emitted per second by 
a sodium lamp rated at 100 W which radiated all its energy 
with 100 per cent eAciency as yellow light of wavelength 
589 nm?
*0.10 When ultraviolet radiation of wavelength 195 nm 
strikes a certain metal surface, electrons are ejected at 
1.23 Mm s!1. Calculate the speed of electrons ejected from 
the same metal surface by radiation of wavelength 255 nm.
0.11 At what wavelength of incident radiation do the 
relativistic and non-relativistic expressions for the ejection 
of electrons from potassium diCer by 10 per cent? That is, 
find l such that the non-relativistic and relativistic linear 
momenta of the photoelectron diCer by 10 per cent. 
Use F = 2.3 eV.
0.12 Deduce eqn 0.10 for the Compton eCect on the basis 
of the conservation of energy and linear momentum. Hint.
Use the relativistic expressions. Initially the electron is at 
rest with energy mec2. When it is travelling with momentum 
p its energy is (p2c2 + m2ec4)1/2. The photon, with initial 
momentum h/li and energy hni, strikes the stationary 
electron, is deflected through an angle q, and emerges 
with momentum h/lf and energy hnf. The electron is initially 
stationary (p = 0) but moves oC with an angle q% to the 
incident photon. Conserve energy and both components 
of linear momentum (parallel and perpendicular to the 
* Indicates that the solution can be found in the Student’s solution manual, which is available in the Online Resource 
Centre accompanying this book. Go to www.oxfordtextbooks.co.uk/orc/mqm5e/
www.oxfordtextbooks.co.uk/orc/mqm5e/
8 | INTRODUCTION AND ORIENTATION
initial momentum). Eliminate q%, then p, and so arrive at 
an expression for dl.
*0.13 The first few lines of the visible (Balmer) series in 
the spectrum of atomic hydrogen lie at l/nm = 656.46, 
486.27, 434.17, 410.29, . . . . Find a value of RH, the 
Rydberg constant for hydrogen. The ionization energy, I,
is the minimum energy required to remove the electron. 
Find it from the data and express its value in electronvolts 
(1 eV = 1.602 " 10!19 J). How is I related to RH? Hint. The 
ionization limit corresponds to n $ & for the final state of 
the electron.
0.14 Use eqn 0.13 for the energy levels of an electron 
in a hydrogen atom to determine an expression for the 
Rydberg constant (as a wavenumber) in terms of 
fundamental constants. Evaluate the Rydberg constant 
(a) using the reduced mass of a hydrogen atom, 
(b) substituting the mass of the electron for the 
reduced mass. (c) What is the percentage diCerence
between the two expressions?
0.15 Derive an expression that could be used to determine 
the mass of a deuteron from the shift in spectral lines of 
1H and 2H.
*0.16 A measure of the strength of coupling between 
the electromagnetic field and an electric charge is the 
fine-structure constant, a = e2/4pHce0. Express the 
Rydberg constant (as a wavenumber) in terms of this 
constant.
0.17 Calculate the de Broglie wavelength of (a) a mass of 
1.0 g travelling at 1.0 cm s!1, (b) the same at 95 per cent of 
the speed of light, (c) a hydrogen atom at room temperature 
(300 K); estimate the mean speed from the equipartition 
principle, which implies that the mean kinetic energy of 
an atom is equal to 32 kT, where k is Boltzmann’s constant, 
(d) an electron accelerated from rest through a potential 
diCerence of (i) 1.0 V, (ii) 10 kV. Hint. For the momentum 
in (b) use p = mv/(l ! v2/c2)1/2 and for the speed in (d) use 
1/2mev2 = eV, where V is the potential diCerence.
0.18 Derive eqn 0.13 for the permitted energy levels for the 
electron in a hydrogen atom. To do so, use the following 
(incorrect) postulates of Bohr: (a) the electron moves in 
a circular orbit of radius r around the nucleus and (b) the 
angular momentum of the electron is an integral multiple 
of H, that is mevr = nH. Hint. Mechanical stability of the 
orbital motion requires that the Coulombic force of 
attraction between the electron and nucleus equals the 
centrifugal force due to the circular motion. The energy of 
the electron is the sum of the kinetic energy and potential 
(Coulombic) energy. For simplicity, use me rather than 
the reduced mass m.
* Indicates that the solution can be found in the Student’s solution manual, which is available in the Online Resource 
Centre accompanying this book. Go to www.oxfordtextbooks.co.uk/orc/mqm5e/
www.oxfordtextbooks.co.uk/orc/mqm5e/
Operators in quantum 
mechanics 9
1.1 Linear operators 10
1.2 Eigenfunctions and 
eigenvalues 10
1.3 Representations 12
1.4 Commutation and 
non-commutation 13
1.5 The construction of 
operators 14
1.6 Integralsover 
operators 15
1.7 Dirac bracket and 
matrix notation 16
1.8 Hermitian operators 17
The postulates of quantum 
mechanics 20
1.9 States and 
wavefunctions 20
1.10 The fundamental 
prescription 21
1.11 The outcome of 
measurements 22
1.12 The interpretation of 
the wavefunction 24
1.13 The equation for the 
wavefunction 24
1.14 The separation of the 
Schrödinger equation 25
The specifi cation and 
evolution of states 26
1.15 Simultaneous 
observables 27
1.16 The uncertainty 
principle 28
1.17 Consequences of the 
uncertainty principle 30
1.18 The uncertainty in 
energy and time 31
1.19 Time-evolution and 
conservation laws 31
1The foundations of quantum mechanics
The whole of quantum mechanics can be expressed in terms of a small set of postu-
lates. When their consequences are developed, they embrace the behaviour of all known 
forms of matter, including the molecules, atoms, and electrons that will be at the centre 
of our attention in this book. This chapter introduces the postulates and illustrates 
how they are used. The remaining chapters build on them, and show how to apply them 
to problems of chemical interest, such as atomic and molecular structure and the 
properties of molecules. We assume that you have already met the concepts of ‘hamil-
tonian’ and ‘wavefunction’ in an elementary introduction, and have seen the Schrödinger 
equation written in the form
Hy = Ey
This chapter establishes the full signifi cance of this equation and provides a foundation 
for its application in the following chapters. It will also be helpful to bear in mind the solu-
tions of the Schrödinger equation for a particle in a box, which we also presume to be 
generally familiar. In brief, for a particle of mass m in a one-dimensional box of length L:
• The energies are quantized, with En = n2h2/8mL2, n = 1,2, . . .
• The normalized wavefunctions are yn(x) = (2/L)1/2 sin(npx/L)
We use these solutions to illustrate some of the points made in this chapter (they are 
developed formally in Chapter 2). A fi nal preparatory point is that quantum mechanics 
makes extensive use of complex numbers: they are reviewed in Mathematical back-
ground 1 following this chapter.
Operators in quantum mechanics
An observable is any dynamical variable that can be measured. The principal 
mathematical diCerence between classical mechanics and quantum mechanics 
is that whereas in the former physical observables are represented by functions 
(such as position as a function of time), in quantum mechanics they are repre-
sented by mathematical operators. An operator is a symbol for an instruction to 
carry out some action, an operation, on a function. In most of the examples we 
shall meet, the action will be nothing more complicated than multiplication or 
diCerentiation. Thus, one typical operation might be multiplication by x, which 
is represented by the operator x ". Another operation might be diCerentiation
with respect to x, represented by the operator d /dx. We shall represent operators 
by the symbol W (uppercase omega) in general, but use A, B, . . . when we want 
to refer to a series of operators. We shall not in general distinguish between the 
observable and the operator that represents that observable; so the position of 
a particle along the x-axis will be denoted x and the corresponding operator will 
also be denoted x (with multiplication implied). We shall always make it clear 
whether we are referring to the observable or the operator.
10 | 1 THE FOUNDATIONS OF QUANTUM MECHANICS
We shall need a number of concepts related to operators and functions on 
which they operate, and this first section introduces some of the more important 
features.
1.1 Linear operators
The operators we shall meet in quantum mechanics are all linear. A linear oper-
ator is one for which
W(af ) = aWf (1.1)
where a is a constant and f is a function. Multiplication is a linear operation; so 
are diCerentiation and integration. An example of a non-linear operation is that 
of taking the logarithm of a function, because it is not true, for example, that 
log 2x = 2 log x for all x. The operation of taking a square is also non-linear, 
because it is not true, for example, that (2x)2 = 2x2 for all x.
1.2 Eigenfunctions and eigenvalues
In general, when an operator operates on a function, the outcome is another 
function. DiCerentiation of sin x, for instance, gives cos x. However, in certain 
cases, the outcome of an operation is the same function multiplied by a constant. 
Functions of this kind are called ‘eigenfunctions’ of the operator. More formally, 
a function f (which may be complex) is an eigenfunction of an operator W if it 
satisfies an equation of the form
Wf = wf (1.2)
where w is a constant. Such an equation is called an eigenvalue equation. The 
function eax is an eigenfunction of the operator d /dx because (d /dx)eax = aeax,
which is a constant (a) multiplying the original function. In contrast, eax2 is not an 
eigenfunction of d /dx, because (d /dx)eax2 = 2axeax2, which is a constant (2a) times 
a diFerent function of x (the function xeax2). The constant w in an eigenvalue 
equation is called the eigenvalue of the operator W.
Example 1.1 Determining if a function is an eigenfunction
Is the wavefunction y1(x) = (2/L)1/2 sin(px/L) of a particle in a box an eigenfunc-
tion of the operator d2/dx2 and, if so, what is the corresponding eigenvalue?
Method Perform the indicated operation on the given function and see if the func-
tion satisfies an eigenvalue equation. Use (d /dx)sinax = acosax and (d /dx)cosax
= !a sinax.
Answer The operation on the function yields
d2y1(x)
dx2
= AC
2
L
D
F
1/2 d2sin(px/L)
dx2
= AC
2
L
D
F
1/2A
C
p
L
D
F
dcos(px/L)
dx
= ! AC
2
L
D
F
1/2A
C
p
L
D
F
2
sin(px/L) = !AC
p
L
D
F
2
y1(x)
and we see that the original function reappears multiplied by a constant, so y1(x)
is an eigenfunction of d2/dx2, and its eigenvalue is !(p/L)2.
Self-test 1.1 Is the function e3x+5 an eigenfunction of the operator d2/dx2 and, if so, 
what is the corresponding eigenvalue?
[Yes; 9]
 1.2 EIGENFUNCTIONS AND EIGENVALUES | 11
An important point is that a general function can be expanded in terms of all 
the eigenfunctions of an operator, a so-called complete set of functions. The func-
tions used to construct a general function are called basis functions. That is, 
if fn is an eigenfunction of an operator W with eigenvalue wn (so Wfn = wnfn), then 
a general function g can be expressed as the linear combination
g = '
n
cnfn (1.3)
where the cn are coeAcients and the sum is over a complete set of basis func-
tions fn. For instance, the straight line g = ax can be recreated over a certain range 
(!L/2 ( x ( L/2) by superimposing an infinite number of sine functions, each of 
which is an eigenfunction of the operator d2/dx2:
g(x) = AC
aL
p
D
F
&
'
n=1
{(!1)n+1/n}sin(2npx/L)
(The formulation and illustration of expressions like this are described in 
Mathematical background 5 following Chapter 10.) The same function may also 
be constructed from an infinite number of exponential functions, which are 
eigenfunctions of d /dx.
The advantage of expressing a general function as a linear combination of 
a set of eigenfunctions is that it allows us to deduce the eCect of an operator on 
a function that is not one of its own eigenfunctions. Thus, the eCect of W on g in 
eqn 1.3, using the property of linearity, is simply
Wg = W'
n
cnfn = '
n
cnWfn = '
n
cnwnfn (1.4)
A special case of these linear combinations is when we have a set of degenerate
eigenfunctions, a set of functions with the same eigenvalue. Thus, suppose that 
f1, f2, . . . , fk are all eigenfunctions of the operator W, and that they all correspond 
to the same eigenvalue w:
Wfn = wfn with n = 1,2, . . . , k (1.5)
Then it is quite easy to show that any linear combination of the functions fn is 
also an eigenfunction of W with the same eigenvalue w. The proof is as follows. 
For an arbitrary linear combinationg of the degenerate set of functions, we can 
write
Wg = W 
k
'
n=1
cnfn =
k
'
n=1
cnW fn =
k
'
n=1
cnw fn = w
k
'
n=1
cnfn = wg (1.6)
This expression has the form of an eigenvalue equation (Wg = wg).
Example 1.2 Demonstrating that a linear combination of degenerate 
eigenfunctions is also an eigenfunction
Show that any linear combination of the complex functions e2ix and e!2ix is an 
eigenfunction of the operator d2/dx2, where i = (!1)1/2.
Method Consider an arbitrary linear combination ae2ix + be!2ix and see if the func-
tion satisfies an eigenvalue equation.
Answer First we demonstrate that e2ix and e!2ix are degenerate eigenfunctions:
d2
dx2
e±2ix = d
dx
(±2ie±2ix) = !4e±2ix ››
12 | 1 THE FOUNDATIONS OF QUANTUM MECHANICS
A further technical point is that from N basis functions it is possible to con-
struct N linearly independent combinations. A set of functions g1, g2, . . . , gN is 
said to be linearly independent if we cannot find a set of constants c1, c2, . . . , cN
(other than the trivial set c1 = c2 = · · · = 0) for which
'
i
cigi = 0 (1.7)
A set of functions that are not linearly independent are said to be linearly depend-
ent. From a set of N linearly independent functions, it is possible to construct 
an infinite number of sets of linearly independent combinations, but each set can 
have no more than N members.
where we have used i2 = !1. Both functions correspond to the same eigenvalue, ! 4.
Then we operate on a linear combination of the functions:
d2
dx2
(ae2ix + be!2ix) = !4(ae2ix + be!2ix)
The linear combination satisfies the eigenvalue equation and has the same eigen-
value (! 4) as do the two exponential functions.
Self-test 1.2 Show that any linear combination of the functions sin(3x) and 
cos(3x) is an eigenfunction of the operator d2/dx2.
[Eigenvalue is !9]
A brief illustration
Consider an H1s orbital on each hydrogen atom in NH3, and denote them sA, sB,
and sC. The three linear combinations
2sA ! sB ! sC 2sB ! sC ! sA 2sC ! sA ! sB
are not linearly independent (their sum is zero). Put another way: the third can be 
expressed as the sum of the first two. On the other hand, the linear combinations
2sA ! sB ! sC sA + sB + sC sB ! sC
are linearly independent, and any one cannot be expressed as a sum or diCerence
of the other two. The three p orbitals (px, py, pz) of a shell of an atom are linearly 
independent. It is possible to form any number of sets of linearly independent 
combinations of them, but each set has no more than three members. One such set 
(which will be discussed further in Section 3.15) is
p+1 = !
1
21/2
(px + ipy) p!1 =
1
21/2
(px ! ipy) p0 = pz
1.3 Representations
The remaining work of this section is to put forward some explicit forms of the 
operators we shall meet. Much of quantum mechanics can be developed in terms 
of an abstract set of operators, as we shall see later. However, it is often fruitful 
to adopt an explicit form for particular operators and to express them in terms 
of the mathematical operations of multiplication, diCerentiation, and so on. 
DiCerent choices of the operators that correspond to a particular observable give 
rise to the diCerent representations of quantum mechanics, because the explicit 
 1.4 COMMUTATION AND NON-COMMUTATION | 13
forms of the operators represent the abstract structure of the theory in terms of 
actual manipulations.
One of the most common representations is the position representation, in 
which the position operator is represented by multiplication by x (or whatever 
coordinate is specified) and the linear momentum parallel to x is represented by 
diCerentiation with respect to x. Explicitly:
Position representation: x $ x " px $ 
H
i
[
[x
(1.8)
where H = h/2#. We replace the partial derivative, [/[x, by an ordinary derivative, 
d /dx, when considering one-dimensional systems in which x is the only variable. 
Why the linear momentum should be represented in precisely this manner is 
explained in the following section. For the time being, it may be taken to be 
a basic postulate of quantum mechanics. An alternative choice of operators is 
the momentum representation, in which the linear momentum parallel to x is 
represented by the operation of multiplication by px and the position operator 
is represented by diCerentiation with respect to px. Explicitly:
Momentum representation: x $ !H
i
[
[px
px $ px " (1.9)
There are other representations. We shall normally use the position representa-
tion when the adoption of a representation is appropriate, but we shall also see 
that many of the calculations in quantum mechanics can be done independently 
of a representation.
1.4 Commutation and non-commutation
An important feature of operators is that in general the outcome of successive 
operations (A followed by B, which is denoted BA, or B followed by A, denoted 
AB) depends on the order in which the operations are carried out. That is, in 
general BA ) AB. We say that, in general, operators do not commute.
A brief illustration
Consider the operators x and px and a specific function x2. In the position 
representation,
(xpx)x2 = x "
H
i
d
dx
x2 = !2iHx2
whereas
(pxx)x2 =
H
i
d
dx
x " x2 = !3iHx2
We see that because the outcomes are diCerent, the operators x and px do not 
commute.
The quantity AB ! BA is called the commutator of A and B and is denoted 
[A,B]:
[A,B] = AB ! BA (1.10)
It is instructive to evaluate the commutator of the position and linear momentum 
operators in the two representations shown above; the procedure is illustrated in 
the following example.
14 | 1 THE FOUNDATIONS OF QUANTUM MECHANICS
The non-commutation of operators is highly reminiscent of the non-commutation 
of matrix multiplication. Indeed, Heisenberg formulated his version of quantum 
mechanics, which is called matrix mechanics, by representing position and linear 
momentum by the matrices x and px, and requiring that xpx ! pxx = iH1 where 1
is the unit matrix, a square matrix with all diagonal elements equal to 1 and 
all others 0. (Matrices are discussed in Mathematical background 4 following 
Chapter 5.)
1.5 The construction of operators
Operators for other observables of interest can be constructed from the operators 
for position and momentum. For example, the kinetic energy operator T can be 
constructed by noting that kinetic energy is related to linear momentum by 
T = p2/2m, where m is the mass of the particle and p2 (in general W2) means that 
the operator is applied twice in succession. It follows that in one dimension and 
in the position representation
T = p
2
x
2m
= 1
2m
A
C
H
i
d
dx
D
F
2
= ! H
2
2m
d2
dx2
 (1.11a)
In three dimensions the operator in the position representation is
T = ! H
2
2m
!
@
[2
[x2
+ [
2
[y2
+ [
2
[z2
#
$ = ! 
H2
2m
*2 (1.11b)
The operator *2, which is read ‘del squared’ and called the laplacian, is the sum 
of the three second derivatives.
Because the potential energy depends only on position coordinates, the oper-
ator for potential energy of a particle in one dimension, V(x), is multiplication by 
the function V(x) in the position representation. The same is true of the potential 
energy operator in three dimensions. For example, in the position representation 
the operator for the Coulomb potential energy of an electron (charge !e) in the 
field of a nucleus of atomic number Z and charge Ze is the multiplicative operator
Example 1.3 Evaluating a commutator
Evaluate the commutator [x,px] in the position representation.
Method To evaluate the commutator [A,B] we need to remember that the oper-
ators operate on some function, which we shall write f. So, evaluate [A,B]f for 
an arbitrary function f, and then cancel f at the end of the calculation.
Answer Substitution of the explicit expressions for the operators into [x,px] pro-
ceeds as follows:
[x,px]f = (xpx ! pxx)f = x "
H
i
[f
[x
! H
i
[(xf )
[x
= x " H
i
[f
[x
! H
i
f ! x " H
i
[f
[x
= iHf
where we have used (1/i) = !i. This derivationis true for any function f, so in terms 
of the operators themselves, [x,px] = iH. The right-hand side of this expression 
should be interpreted as the operator ‘multiply by the constant iH’.
Self-test 1.3 Evaluate the same commutator in the momentum representation.
[Same]
A brief comment
Although eqn 1.11b has 
explicitly used Cartesian 
coordinates, the relation 
between the kinetic energy 
operator and the laplacian is 
true in any coordinate system; 
for example, spherical polar 
coordinates. These alternative 
versions of the laplacian are 
given in Mathematical
background 3 following 
Chapter 4.
V = ! Ze
2
4pe0r
(1.12)
where r is the distance from the nucleus to the electron. As here, it is usual to 
omit the multiplication sign from multiplicative operators, but it should not be 
forgotten that such expressions imply multiplications of whatever stands on their 
right.
The operator for the total energy of a system is called the hamiltonian operator
and is denoted H:
H = T + V (1.13)
The name commemorates W.R. Hamilton’s contribution to the formulation of 
classical mechanics in terms of what became known as a hamiltonian function.
To write the explicit form of this operator we simply substitute the appropriate 
expressions for the kinetic and potential energy operators in the chosen represen-
tation. For example, the hamiltonian operator for a particle of mass m moving in 
one dimension is
H = ! H
2
2m
d2
dx2
+ V(x) (1.14)
where V(x) is the operator for the potential energy. Similarly, the hamiltonian 
operator (from now on, just ‘the hamiltonian’) for an electron of mass me in 
a hydrogen atom is
H = ! H
2
2me
 *2 ! e
2
4pe0r
(1.15)
The general prescription for constructing operators in the position representa-
tion should be clear from these examples. In short:
1. Write the classical expression for the observable in terms of position co-
ordinates and the linear momentum.
2. Replace x by multiplication by x, and replace px by (H/i)[/[x (and likewise 
for the other coordinates).
1.6 Integrals over operators
When we want to make contact between a calculation done using operators and 
the actual outcome of an experiment, it will turn out that we shall need to evalu-
ate certain integrals. These integrals all have the form
I = ! f*mWfn dt (1.16)
where f*m is the complex conjugate (Mathematical background 1) of fm. In this 
integral dt is the volume element. In one dimension, dt can be identified as dx; in 
three dimensions it is dxdydz. The integral is taken over the entire space avail-
able to the system, which is typically from x = !& to x = +& (and similarly for the 
other coordinates). A glance at the later pages of this book will show that many 
molecular properties are expressed as combinations of integrals of this form 
(often in a notation which will be explained later). Certain special cases of this 
type of integral have special names, and we shall introduce them here.
When the operator W in eqn 1.16 is simply multiplication by 1, the integral is 
called an overlap integral and commonly denoted S:
S = ! f*mfn dt (1.17)
 1.6 INTEGRALS OVER OPERATORS | 15
16 | 1 THE FOUNDATIONS OF QUANTUM MECHANICS
It is helpful to regard S as a measure of the similarity of two functions: when 
S = 0, the functions are classified as orthogonal, rather like two perpendicular 
vectors. When S is close to 1, the two functions are almost identical. The recogni-
tion of mutually orthogonal functions often helps to reduce the amount of calcu-
lation considerably, and rules will emerge in later sections and chapters.
The normalization integral is the special case of eqn 1.17 for m = n. A function 
fm is said to be normalized (strictly, normalized to 1) if
! f*mfm dt = 1 (1.18)
The integration here, as (by convention) it always is when dt is used to denote 
the volume element, is over all space. It is almost always easy to ensure that 
a function is normalized by multiplying it by an appropriate numerical factor, 
which is called a normalization factor, typically denoted N and taken to be real 
so that N* = N. We could take N to have any complex phase, but because all 
observables are proportional to N*N, the phase cancels and it is simply conveni-
ent to make N real. The procedure is illustrated in the following example.
Example 1.4 Normalizing a function
The ground state wavefunction of a particle in a box is y1(x) = N sin(px/L) between
x = 0 and x = L and is zero elsewhere. Confirm that N = (2/L)1/2.
Method To find N we substitute this expression into eqn 1.18, evaluate the inte-
gral, and select N to ensure normalization. Note that ‘all space’ in eCect extends 
from x = 0 to x = L because the function is identically zero outside this region.
Answer The necessary integration is
! f*fdt = !
L
0 
N2 sin2(px/L)dx = 12LN2
where we have used + sin2 ax dx = (x/2) ! (sin 2ax)/4a + constant. For this integral 
to be equal to 1, we require N = (2/L)1/2.
Self-test 1.4 Normalize the function f = eij, where j ranges from 0 to 2p.
[N = 1/(2p)1/2]
A set of functions fn that are (a) normalized and (b) mutually orthogonal are 
said to satisfy the orthonormality condition:
! f*mfn dt = dmn (1.19)
In this expression, dmn denotes the Kronecker delta, which is 1 when m = n and 
0 otherwise.
1.7 Dirac bracket and matrix notation
The appearance of many quantum mechanical expressions is greatly simplified 
by adopting a simplified notation.
(a) Dirac brackets
In the Dirac bracket notation integrals are written as follows:
,m | W | n- = ! f*mWfn dt (1.20)
 1.8 HERMITIAN OPERATORS | 17
The symbol | n- is called a ket, and denotes the state described by the function fn.
Similarly, the symbol ,n | is called a bra, and denotes the complex conjugate of the 
function, f*n. When a bra and ket are strung together with an operator between 
them, as in the bracket ,m |W |n-, the integral in eqn 1.20 is to be understood. 
When the operator is simply multiplication by 1, the 1 is omitted and we use the 
convention
,m | n- = ! f*mfn dt (1.21)
This notation is very elegant. For example, the normalization integral becomes 
,n |n- = 1 and the orthogonality condition becomes ,m |n- = 0 for m ) n. The 
combined orthonormality condition (eqn 1.19) is then
,m |n- = dmn (1.22)
A further point is that, as can readily be deduced from the definition of a Dirac 
bracket,
,m |n- = ,n |m-* (1.23)
(b) Matrix notation
A matrix, M, is an array of numbers (which may be complex), called matrix
elements. Each element is specified by quoting the row (r) and column (c) that it 
occupies, and denoting the matrix element as Mrc. The rules of matrix algebra are 
set out in Mathematical background 4 following Chapter 5, where they are centre 
stage. Dirac brackets are commonly abbreviated to Wmn, which immediately 
suggests that they are elements of a matrix. For this reason, the Dirac bracket 
,m |W |n- is often called a matrix element of the operator W. A diagonal matrix 
element Wnn is then a bracket of the form ,n |W |n- with the bra and the ket refer-
ring to the same state.
We shall often encounter sums over products of Dirac brackets that have the 
form
'
s
,r |A | s-,s |B |c-
If the brackets that appear in this expression are interpreted as matrix elements, 
then we see that it has the form of a matrix multiplication, and we may write
'
s
,r |A | s-,s |B |c- = '
s
ArsBsc = (AB)rc = ,r |AB |c- (1.24)
That is, the sum is equal to the single matrix element (bracket) of the product of 
operators AB. Comparison of the first and last terms in this line of equations also 
allows us to write the symbolic relation
'
s
| s-,s | = 1 (1.25)
This completeness relation (or closure relation) is exceptionally useful for devel-
oping quantum mechanical equations. It is often used in reverse: the matrix 
element ,r |AB |c- can always be split into a sum of two factors by regarding it 
as ,r |A1B |c- and then replacing the 1 by a sum over a complete set of states of 
the form in eqn 1.25.
1.8 Hermitian operators